Area and Perimeter of Different Types of Triangle

Perimeter of triangle

The perimeter is the length of the boundary around any closed figure.
The boundary of a triangle consists of its three sides. Therefore, the sum of the length of three sides of a triangle is called its perimeter.

Perimeter of triangle ΔABC

In ΔABC,
length of AB = a
length of BC = b
and length of CA = c
So, perimeter of ΔABC is sum of length of sides AB, BC and CA
i.e. perimeter of ΔABC = a + b + c

What are the units of perimeter?

Units of perimeter measurement of a triangle are taken the same as units of length of sides of the triangle.
The few examples of units of length on the side of a triangle are meter (m), centimeter (cm) etc.. So, the units of perimeter are also taken as meter (m), centimeter (cm) etc..

Perimeter of an equilateral triangle

An equilateral triangle has all of its sides of equal length.
∴ Perimeter of an equilateral triangle can be calculated by adding up its all three sides or by multiplying length of its side with 3.

Equilateral triangle ΔABC with sides length a

In equilateral triangle ΔABC,
AB = a, BC = a and CA = a
So, perimeter of equilateral ΔABC = a + a + a = 3a

Perimeter of isosceles triangle

In an isosceles triangle, any two sides are of equal length.

Isosceles triangle ΔABC with sides length a, b and a

AB = a, BC = a and CA = b
So, perimeter of isosceles ΔABC = a + a + b = 2a + b

Area of triangle

Area is the total amount of space occupied by any closed figure.

Area of any triangle = $\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$

Area of triangle ΔABC

In ΔABC, BC is base, length of BC = b
and AO is height, length of AO = h
Therefore, area of ΔABC = $\frac{1}{2}\times b\times h$

What are the units of area?

Area is measured in square units.
The few examples of units of area of a triangle are meter² or m² read as meter square, centimeter² or cm² read as centimeter square etc.

Area of equilateral triangle

In equilateral, where all sides of triangle are equal in length, its area is calculated as:
Area of ΔABC = $\frac{\sqrt{3}}{4}{\mathrm{a}}^2$
where a is the length of side
Let's see how it is calculated?
Here, in right angle ΔAOC
$a^2=h^2+{\left(\frac{a}{2}\right)}^2$       by Pythagoras theorem
$a^2=h^2+\frac{a^2}{4}$
$a^2-\frac{a^2}{4}=h^2$
$\frac{4a^2-a^2}{4}=h^2$
$\frac{3a^2}{4}=h^2$
$\frac{\sqrt{3}a}{2}=h$
Area of equilateral Δ ABC = $\frac{1}{2}\times a\times \frac{\sqrt{3}a}{2}$
= $\frac{\sqrt{3}}{4}{\mathrm{a}}^2$

Area of isosceles triangle

area of isosceles ΔABC = $\frac{b\sqrt{{\mathrm{4a}}^2-b^2}}{4}$
Let's see how it is calculated?
Here, in right angle ΔAOC
$a^2=h^2+{\left(\frac{b}{2}\right)}^2$           by Pythagoras theorem
$a^2=h^2+\frac{b^2}{4}$
$a^2-\frac{b^2}{4}=h^2$
$\frac{{\mathrm{4a}}^2-b^2}{4}=h$
$\frac{\sqrt{{\mathrm{4a}}^2-b^2}}{2}=h$
we know, area of Δ = $\frac{1}{2}\times b\times h$
∴ area of isosceles ΔABC = $\frac{1}{2}\times b\times \frac{\sqrt{{\mathrm{4a}}^2-b^2}}{2}$
= $\frac{b\sqrt{{\mathrm{4a}}^2-b^2}}{4}$